Is Z6 a field? This question has intrigued mathematicians and students of abstract algebra for years. In this article, we will explore the concept of a field and delve into whether Z6, the set of integers modulo 6, meets the criteria to be classified as a field.
Fields are a fundamental concept in abstract algebra, representing a mathematical structure that is similar to the set of rational numbers, but with some key differences. A field is a set equipped with two operations, addition and multiplication, that satisfy certain properties. These properties include commutativity, associativity, the existence of an additive and a multiplicative identity, and the existence of additive and multiplicative inverses for all non-zero elements.
To determine whether Z6 is a field, we need to examine whether it satisfies these properties. The set Z6 consists of the integers 0, 1, 2, 3, 4, and 5, with the operations of addition and multiplication performed modulo 6. In other words, when we add or multiply two integers in Z6, we take the remainder after dividing by 6.
Let’s begin by checking the commutative and associative properties. It is straightforward to verify that both addition and multiplication in Z6 are commutative and associative. For example, 1 + 2 = 2 + 1 = 3, and (1 + 2) + 3 = 1 + (2 + 3) = 6 ≡ 0 (mod 6). Similarly, 2 3 = 3 2 = 6 ≡ 0 (mod 6), and (2 3) 4 = 2 (3 4) = 24 ≡ 0 (mod 6).
Next, we need to confirm that Z6 has an additive and a multiplicative identity. The additive identity is 0, as 0 + a = a + 0 = a for any integer a. The multiplicative identity is 1, as 1 a = a 1 = a for any integer a.
Finally, we must establish that every non-zero element in Z6 has an additive and a multiplicative inverse. The additive inverse of an integer a is the integer -a, which satisfies a + (-a) ≡ 0 (mod 6). For example, the additive inverse of 2 is 4, since 2 + 4 = 6 ≡ 0 (mod 6). The multiplicative inverse of an integer a is the integer b such that a b ≡ 1 (mod 6). For instance, the multiplicative inverse of 2 is 3, as 2 3 = 6 ≡ 0 (mod 6).
However, there is a problem with Z6 that prevents it from being a field. In a field, every non-zero element must have a multiplicative inverse. But in Z6, the element 2 does not have a multiplicative inverse. This is because there is no integer b such that 2 b ≡ 1 (mod 6). To see this, we can try multiplying 2 by each element of Z6 and check if the result is congruent to 1 modulo 6:
2 0 ≡ 0 (mod 6)
2 1 ≡ 2 (mod 6)
2 2 ≡ 4 (mod 6)
2 3 ≡ 0 (mod 6)
2 4 ≡ 2 (mod 6)
2 5 ≡ 2 (mod 6)
As we can see, no element in Z6, when multiplied by 2, gives a result congruent to 1 modulo 6. Therefore, Z6 does not satisfy the requirement for a field, and the answer to the question “Is Z6 a field?” is no.
